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RT is staggered by the mysteries of the mind. He read Euclid in college, but he is struggling to understand how a colony of bees can produce a structure as complicated as a honeycomb. And not only complicated, but efficient as well. Do honeybees compose elaborate poems as they work away? Some of the honeycomb maker’s art informs the efforts of the poet; maybe Yeats was onto something in the “Lake Isle of Innishfree.”

In any case, here is the description of a honeycomb from the Wikipedia Page:

“The axes of honeycomb cells are always quasi-horizontal, and the non-angled rows of honeycomb cells are always horizontally (not vertically) aligned. Thus, each cell has two vertical walls, with “floors” and “ceilings” composed of two angled walls. The cells slope slightly upwards, between 9 and 14 degrees, towards the open ends.

There are two possible explanations for the reason that honeycomb is composed of hexagons, rather than any other shape. One, given by Jan Brożek and proved much later by Thomas Hales, is that the hexagon tiles the plane with minimal surface area. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. Another, given by D’Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency.

The closed ends of the honeycomb cells are also an example of geometric efficiency, albeit three-dimensional and little-noticed. The ends are trihedral (i.e., composed of three planes) sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume. (The angle formed by the edges at the pyramidal apex, known as the tetrahedral angle, is approximately 109° 28′ 16″ (= arccos(−1/3)).)”

Pre-columbian art, connected as it is with blood worship, has a reputation for being a little fantastical and spooky. While RT readily acknowledges that some of the religious art produced by pre-columbian civilizations can be hair-raising, he also thinks there is a playful, even whimsical, side to their craft work. Here we have an “amphibian” pendant from Panama. Putting aside the “flippers” or tail or whatever they are, RT thinks there is something decidedly frog-like about the beast. There is a deep well of invention and creativity in the native peoples of the Americas, something that may re-emerge to enrich our lives.